Overview¶

Maxwell's equations¶

Maxwell's equations are a system of equations that describe the electromagnetic field and its connection with electric charges and currents. Maxwell's equations in matter can be written as follows:

\[ \begin{align} \nabla \cdot \mathbf{D} &= \rho, && \qquad \textrm{(Gauss's law)} \\ \nabla \cdot \mathbf{B} &= 0, && \qquad \textrm{(Gauss's law for magnetism)}\\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, && \qquad \textrm{(Faraday's law)}\\ \nabla \times \mathbf{H} &= \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}, && \qquad \textrm{(Ampère's law with Maxwell's correction)} \end{align} \]

where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, \(\mathbf{D}\) is the electric displacement field, and \(\mathbf{H}\) is the magnetic field, with \(\rho\) being the density of free electric charges and \(\mathbf{J}\) being the electric current density.

The response of the medium to the electromagnetic field is described by the following constitutive relations:

\[ \begin{align} \mathbf{D} &= \varepsilon_0 \mathbf{E} + \mathbf{P}_e\, , \\ \mathbf{B} &= \mu_0 \mathbf{H} + \mathbf{P}_m\, , \end{align} \]

where \(\varepsilon_0\) is the vacuum permittivity, \(\mu_0\) is the vacuum permeability, \(\mathbf{P}_e\) is the electric polarization and \(\mathbf{P}_m\) is the magnetic polarization. Note that quite often instead of the magnetic polarization \(\mathbf{P}_m\) one uses the magnetization \(\mathbf{M}\) defined as \(\mathbf{M}=\mathbf{P}_m/\mu_0\).

For media whose response does not depend on the field strength (linear media) and on the frequency of the electromagnetic field (non-dispersive media), the electric and magnetic polarization can be written as \(\mathbf{P}_e = \varepsilon_0 \chi_e \mathbf{E}\) and \(\mathbf{P}_m = \mu_0 \chi_m \mathbf{H}\), where \(\chi_e\) and \(\chi_m\) are the electric and magnetic susceptibilities. Note that in anisotropic media, where the response of the medium depends on the direction of the electromagnetic field, the electric and magnetic susceptibilities are tensors, and the product of \(\chi_e\) or \(\chi_m\) with the corresponding field vector should be understood as a tensor product. Using these expressions for the polarizations we can rewrite the constitutive relations as follows:

\[ \begin{align} \mathbf{D} &= \varepsilon_0\varepsilon_r \mathbf{E}, \\ \mathbf{B} &= \mu_0\mu_r \mathbf{H}, \end{align} \]

where \(\varepsilon_r = 1 + \chi_e\) is the medium relative permittivity and \(\mu_r = 1 + \chi_m\) is the medium relative permeability.

Time-harmonic fields¶

Any time-dependent electromagnetic field can be represented as a superposition of harmonic waves, each of which oscillates at its own frequency. Mathematically, this statement can be expressed as the corresponding Fourier transform of the field vector function, which, using the example of the electric field \(\mathbf{E}\), can be written as

\[ \begin{align} \mathbf{E}(\mathbf{r}, t) = \int_{-\infty}^{\infty} \tilde{\mathbf{E}}(\mathbf{r}, \omega)\, e^{j\omega t}\, d\omega, \end{align} \]

where \(\tilde{\mathbf{E}}(\mathbf{r}, \omega)\) is the complex spectral amplitude of a harmonic wave \(e^{j\omega t}\) oscillating at the frequency \(f=\omega/2\pi\), with \(\omega\) being the angular frequency and \(\mathbf{r}\) being the spatial coordinate vector.

Taking into account that the spectral amplitude of the time derivative of a function is equal to the spectral amplitude of the function itself, multiplied by \(j\omega\), we can rewrite Maxwell's equations in terms of complex spectral amplitudes as follows:

\[ \begin{align} \mathbf{\nabla} \cdot \tilde{\mathbf{D}} &= \tilde{\rho}, \\ \nabla \cdot \tilde{\mathbf{B}} &= 0, \\ \nabla \times \tilde{\mathbf{E}} &= -j\omega \tilde{\mathbf{B}}, \\ \nabla \times \tilde{\mathbf{H}} &= \tilde{\mathbf{J}} +j\omega \tilde{\mathbf{D}}, \end{align} \]

where the tilde symbol denotes the complex spectral amplitude of the corresponding vector function.

Similar to the electric field vectors, the constitutive relations for the spectral amplitudes can be written as

\[ \begin{align} \tilde{\mathbf{D}}(\mathbf{r},\omega) &= \varepsilon(\mathbf{r},\omega)\, \tilde{\mathbf{E}}(\mathbf{r},\omega), \\ \tilde{\mathbf{B}}(\mathbf{r},\omega) &= \mu(\mathbf{r},\omega)\, \tilde{\mathbf{H}}(\mathbf{r},\omega), \end{align} \]

where \(\varepsilon\) and \(\mu\) are the complex permittivity and permeability of the medium at a given frequency \(\omega\). The explicit dependence of the permittivity and permeability on the frequency in these expressions allows us to consider the dispersive media whose response depends on the frequency of the electromagnetic field.

Electromagnetic Models¶

The choice of model depends on the physics of the application:

Application	Model
Static voltages and capacitances — HV insulator design, capacitor / MEMS comb-drive characterisation, Faraday-cage shielding, semiconductor space charge, electron-optics	Electrostatics
Permanent-magnet machines, solenoid actuators, magnetic bearings, transient excitation, cogging-force and -torque studies, eddy-current transients	Time-Domain Magnetic
Steady-state AC eddy currents — induction heating of conductive workpieces, skin- and proximity-effect losses in bus-bars and windings, transformer end-region analysis, AC equivalent-circuit extraction	Time-Harmonic Magnetic
Wave propagation — antennas, waveguides, resonant cavities, microwave components, electromagnetic-compatibility (EMC), scattering	Time-Harmonic Maxwell

The remaining subsections sketch the formulation each model solves; follow the link in the table for the full per-model documentation.

Electrostatics¶

Used when only static electric fields are of interest. For a static field \(\nabla \times \mathbf{E} = 0\), so \(\mathbf{E} = -\nabla \phi\) and Gauss's law reduces to Poisson's equation:

\[ \nabla \cdot \left( \varepsilon \nabla \phi \right) = -\rho. \]

Typical setups combine prescribed-voltage electrodes (Dirichlet), surface- or volume-charge sources (Neumann), and floating-potential conductors whose equipotential value is part of the solution. See the Electrostatics Model page for the full description.

Time-Domain Magnetic¶

Used when magnetic fields and eddy currents dominate and the device is much smaller than a wavelength. The displacement current is dropped (quasi-static approximation), the magnetic flux density is written as \(\mathbf{B} = \nabla \times \mathbf{A}\), and Ohm's law closes the system. The model integrates

\[ \nabla \times \left( \nu \, \nabla \times \mathbf{A} \right) + \sigma \frac{\partial \mathbf{A}}{\partial t} = \mathbf{J}_\mathrm{s} + \nabla \times \left( \nu \mathbf{B}_r \right) \]

in time, where \(\nu = 1/\mu\) is the magnetic reluctivity and \(\mathbf{B}_r\) accounts for permanent magnets. Transient excitation, non-linear iron, and remanent magnetisation are all handled directly. See the Time-Domain Magnetic Model page for the full description.

Time-Harmonic Magnetic¶

The frequency-domain counterpart of the time-domain magnetic model: when the excitation is sinusoidal at a single frequency and the materials are (effectively) linear, the time derivative collapses to \(j\omega\) and the system reduces to a complex-amplitude problem:

\[ \nabla \times \left( \nu \, \nabla \times \tilde{\mathbf{A}} \right) + j \omega \sigma \tilde{\mathbf{A}} = \tilde{\mathbf{J}}_\mathrm{s}. \]

Use this instead of stepping the time-domain model through many cycles when only the steady-state AC response matters. See the Time-Harmonic Magnetic Model page for the full description.

Time-Harmonic Maxwell¶

Used when the wavelength becomes comparable to the device — the quasi-static approximation breaks down and wave propagation must be resolved. Eliminating \(\tilde{\mathbf{H}}\) from the two harmonic curl equations gives the vector wave equation for the electric field:

\[ \nabla \times \left( \frac{1}{\mu} \nabla \times \tilde{\mathbf{E}} \right) - \omega^2 \varepsilon \, \tilde{\mathbf{E}} = -j \omega \tilde{\mathbf{J}}. \]

See the Time-Harmonic Maxwell Model page for the full description.